1.2 Spatial Coherence of Rhodamine 6G Laser Radiation …
9
E ∼ E 0 exp
−
| r − −
ct|
2
ra
exp[i(k 0 r − ωt)].
(1.8)
From here, it is easily seen that coherence characteristic time of this radiation is
τ =
a
c
, where c is the speed of light, and the coherence length is l = a.
Fourier representation of such a wave package is given by:
E(k) ∼ exp
(k − k 0 )
2
2σ
exp[i(kr − ωt)],
(1.9)
where σ =
1
a
.
Thus, the package under consideration is characterized by wave vector spread k
≈ σ =
1
a
, i.e. kl ≈ 1; similarly for frequency spread we have ωτ ≈ 1.
It can be easily seen from here that for the spatial coherence raise, it is necessary
to reduce the half-width of the radiation line ω. In this context, determination of the
degree of temporal coherence is connected with radiation spectral width measuring.
For the temporal coherence quantitative description, it is convenient to use the notion
of radiation coherence length, which, as it is known [62], can be given by:
=
λ
2
λ
,
(1.10)
where λ is the radiation spectrum width.
For the spatial coherence degree measuring more often interference [63–66],
diffraction [67, 68] and holographic [69–73] methods are used. For the wave fields
characteristic when spatial coherence degree is revealed as spatially inhomogeneous
function [72], it is necessary to know the full spatial coherence function, which
includes mutual-coherence degree between all field points. This problem can be
solved with the help of holographic methods [69, 73].
One of the most effective methods for the spatial coherence studies of dye lasers
is the holographic method proposed by Yu. N. Denisyuk, D. I. Staselko and R. R.
Gerke. The main idea of the holographic method [69] is as follows. A hologram
is a photosensitive layer where the photographic record of the interference pattern
is carried out and which appears under the superposition of reference and object
waves. The contrast of this interference pattern is provided by the radiation spatial–
temporal characteristics of the used source of light. In turn, the brightness of the image
reconstructed by a hologram is in proportion to the interference structure contrast
square. Thus, the brightness of the image reconstructed by a hologram contains radiation spatial–temporal coherence information. The scheme of hologram recording
is diverse, and it depends on the temporal coherence studies [74] or on the spatial
coherence studies of the radiation [42–44, 69].
For the spatial coherence studies, it was proposed [69] to record a hologram in
such a manner that the laser edge is designed on a hologram (reference beam) and on
the flat diffuse screen (object beam). If radiation contains several transverse modes
9
E ∼ E 0 exp
−
| r − −
ct|
2
ra
exp[i(k 0 r − ωt)].
(1.8)
From here, it is easily seen that coherence characteristic time of this radiation is
τ =
a
c
, where c is the speed of light, and the coherence length is l = a.
Fourier representation of such a wave package is given by:
E(k) ∼ exp
(k − k 0 )
2
2σ
exp[i(kr − ωt)],
(1.9)
where σ =
1
a
.
Thus, the package under consideration is characterized by wave vector spread k
≈ σ =
1
a
, i.e. kl ≈ 1; similarly for frequency spread we have ωτ ≈ 1.
It can be easily seen from here that for the spatial coherence raise, it is necessary
to reduce the half-width of the radiation line ω. In this context, determination of the
degree of temporal coherence is connected with radiation spectral width measuring.
For the temporal coherence quantitative description, it is convenient to use the notion
of radiation coherence length, which, as it is known [62], can be given by:
=
λ
2
λ
,
(1.10)
where λ is the radiation spectrum width.
For the spatial coherence degree measuring more often interference [63–66],
diffraction [67, 68] and holographic [69–73] methods are used. For the wave fields
characteristic when spatial coherence degree is revealed as spatially inhomogeneous
function [72], it is necessary to know the full spatial coherence function, which
includes mutual-coherence degree between all field points. This problem can be
solved with the help of holographic methods [69, 73].
One of the most effective methods for the spatial coherence studies of dye lasers
is the holographic method proposed by Yu. N. Denisyuk, D. I. Staselko and R. R.
Gerke. The main idea of the holographic method [69] is as follows. A hologram
is a photosensitive layer where the photographic record of the interference pattern
is carried out and which appears under the superposition of reference and object
waves. The contrast of this interference pattern is provided by the radiation spatial–
temporal characteristics of the used source of light. In turn, the brightness of the image
reconstructed by a hologram is in proportion to the interference structure contrast
square. Thus, the brightness of the image reconstructed by a hologram contains radiation spatial–temporal coherence information. The scheme of hologram recording
is diverse, and it depends on the temporal coherence studies [74] or on the spatial
coherence studies of the radiation [42–44, 69].
For the spatial coherence studies, it was proposed [69] to record a hologram in
such a manner that the laser edge is designed on a hologram (reference beam) and on
the flat diffuse screen (object beam). If radiation contains several transverse modes